超弦理論（第2卷）

Preface

8One-loop diagrams in the bosonic string theory
8.1Open-string one-loop amplitudes
8.1.1 The planar diagrams
8.1.2 The nonorientable diagrams
8.1.3 Nonplanar loop diagrams
8.2Closed-string one-loop amplitudes
8.2.1 The torus
8.2.2 Modular invariance
8.2.3 The integration region
8.2.4 Analysis of divergences
8.2.5 The cosmological constant
8.2.6 Amplitudes with closed-string massless states
8.3 Other diagrams for unoriented strings
8.3.1 Higher-order tree diagrams
8.3.2 The real projective plane
8.3.3 Other loop diagrams
8.4Summary
8.A Jacobi 0 functions

9One-laop diagrams in superstring theory
9.1Opemsuperstring amplitudes
9.1.1 Amplitudes with M 4 massless external states
9.1.2 The planar diagrams
9.1.3 Nonorientable diagrams
9.1.4 Orientable nonplanar diagrams
9.2Type II theories
9.2.1 a Finiteness of the torus amplitude
9.2.2 Compactification on a torus
9.2.3 The low-energy limit of one-loop amplitudes
9.3The hea;erotic string theory
9.3.1 The torus with four external particles
9.3.2 Modular invariance of the Es a Es and SO(32) theories
9.4 Calculations in the RNS formalism
9.4.1 Modular invariance and the GSO projection
9.4.2 The loop calculations
9.5 Orbifolds and twisted strings
9.5.1 Generalization of the GSO projection
9.5.2 Strings on orbifolds
9.5.3 Twisted strings in ten dimensions
9.5.4 Alternative view of the SO(16) a SO(16) theory
9.6 Summary
9.A Traces of fermionic zero modes
9.B Modular invariance of the functions F2 and/:

10The gauge anomaly in type I superstring theory
10.1 Introduction to anomalies
10.1.1 Anomalies in point-particle field theory
10.1.2 The gauge anomaly in D = 10 super
Yang-Mills theory
10.1.3 Anom
10.2.1 The planar diagram anomaly
10.2.2 The anomaly in the nonorientable diagram
10.2.3 Absence of anomalies in nonplanar diagrams
10.3 Other one-loop anomalies in superstring theory
10.4 Cancellation of divergences for SO(32)
10.4.1 Dilaton tadpoles and loop divergences
10.4.2 Divergence cancellations
10.5 Summary
10.A An alternative regulator

11Functional methods in the light-cone gauge
11.1 The string path integral
11.1.1 The analog model
11.1.2 The free string propagator
11.1.3 A lattice cutoff
11.1.4 The continuum limit
11.2 Amplitude calculations
11.2.1 Interaction vertices
11.2.2 Parametrization of scattering processes
11.2.3 Evaluation of the functional integral
11.2.4 Amplitudes with external ground statm
11.3 Open-string tree amplitudes
11.3.1 The conformal mapping
11.3.2 Evaluation of amplitudes
11.4 Open-string trees with excited external states
11.4.1 The Green function on an infinite strip
11.4.2 Green functions for arbitrary tree amplitudes
11.4.3 The amplitude in terms of oscillators
11.4.4 The general form of the Neumann coefficients
11.4.5 The Neumann coefficients for
the cubic open-string vertex
11.5 One-loop open-string amplitudes
11.5.1 The conformal mapping for the planar loop diagram
11.5.2 The Green function
11.5.3 The planar one-loop amplitude
11.5.4 Other one-loop amplitudes
11.6 Closed-string amplitudes
11.6.1 Tree amplitudes
11.6.2 Closed-string one-loop amplitudes
11.7 Superstrings
11.7.1 The SU(4) a U(1) formalism
11.7.2 The super-Poincara generators
11.7.3 Supersymmetry algebra in the interacting theory
11.7.4 The continuity delta functional
11.7.5 Singular operators near the interaction point
11.7.6 The interaction terms
11.7.7 Tree amplitudes for open superstrings
11.8 Summary
11.A The determinant of the Laplacian
11.B The Jacobian for the conformal transformation
11.C Properties of the functions f
11.D Properties of the SU(4) Clebsch-Gordan coefficients

12Some differential geometry
12.1 Spinors in general relativity
12.2 Spin structures on the string world sheet
12.3 Topologically nontrivial gauge fields
12.3.1 The tangent bundle
12.3.2 Gauge fields and vector bundles
12.4 Differential forms
12.5 Characteristic classes
12.5.1 The nonabelian case
12.5.2 Characteristic closes of manifolds
12.5.3 The Euler characteristic:of a Riemann'arface

13Low-energy effective action
13.1 Minimal supergravity plus super Yang-Mills
13.1.1 N - 1 supergravity in ten and eleven dimensions
13.1.2 Type IIB supergravity
13.1.3 The coupled supergravity super Yang-Mills system
13.2 Scale invariance of the classical theory
13.3 Anomaly analysis
13.3.1 Structure of field theory anomalies
13.3.2 Gravitational anomalies
13.3.3 Mixed anomalies
13.3.4 The anomalous Feynman diagrams
13.3.5 Mathematical characterization of anomalies
13.3.6 Other types of anomalies
13.4 Explicit formulas for the anomalies
13.5 Anomaly cancellations
13.5.1 Type I supergravity without matter
13.5.2 Type IIB supergravity
13.5.3 Allowed gauge groups for N -- 1 superstring theories
13.5.4 The SO(16) x SO(16) theory

14Compactiflcation of higher dimensions
14.1 Wave operators in ten dimensions
14.1.1 Massless fields in ten dimensions
14.1.2 Zero modes of wave operators
14.2 Massless fermions
14.2.1 The index of the Dirac operator
14.2.2 Incorporation of gauge fields
14.2.3 The chiral asymmetry
14.2.4 The Parita-Schwinger operator
14.2.5 Outlook
14.3 Zero modes of antisymmetric tensor fields
14.3.1 Antisymmetric tensor fields
14.3.2 Application to axions in N = 1 superstring theory
14.3.3 The 'nonzero modes'
14.3.4 The exterior derivative and the Dirac operator
14.4 Index theorems on the string world sheet
14.4.1 The Dirac index
14.4.2 The Euler characteristic
14.4.3 Zero
14.4.4 Zero modes of superconformal ghosts
14.5 Zero modes of nonlinear fields
14.6 Models of the fermion quantum numbers
14.7 Anomaly cancellation in four dimensions

15Some algebraic geometry
15.1 Low-energy supersymmetry
15.1.1 Motivation
15.1.2 Conditions for unbroken supersymmetry
15.1.3 Manifolds of SU(3) holonomy
15.2 Complex manifolds
15.2.1 Almost complex structure
15.2.2 The Nijenhuis tensor
15.2.3 Examples of complex manifolds
15.3 KS hler manifolds
15.3.1 The Kahler metric
15.3.2 Exterior derivatives
15.3.3 The affine connection and the Riemann tensor
15.3.4 Examples of Kahler manifolds
15.4 Ricci-flat Kahler manifolds and SU(N) halonomy
15.4.1 The Calabi-Yau metric
15.4.2 Covariantly constant forms
15.4.3 Some manifolds of SU(N) holonomy
15.5 Wave operators on Kahler manifolds
15.5.1 The Dirac operator
15.5.2 Dolbeault cohomology
15.5.3 The Hodge decomposition
15.5.4 Hodge numbers
15.6 Yang-Mills equations and holomorphic vector bundles
15.6.1 Holomorphic vector bundles
15.6.2 The Donaldson-Uhlenbeck-Yau equation
15.6.3 Examples
15.7 Dolbeault cohomology and some applications
15.7.1 Zero modes of the Dirac operator
15.7.2 Deformations of complex manifolds
15.7.3 Deformations of holomorphic vector bundles
15.8 Branched coverings of complex manifolds

16Models of low-energy supersymmetry
16.1 A'simple Ansatz
16.2 The spectrum of massless particles
16.2.1 Zero modes of charged fields
16.2.2 Fluctuations of the gravitational field
16.2.3 The other Bose fields
16.3 Symmetry breaking by Wilson lines
16.3.1 Symmetry breaking patterns
16.3.2 A four generation model
16.4 Relation to conventional grand unification
16.4.1 Alternative description of symmetry breaking
16.4.2 Ee relations among coupling constants
16.4.3 Counting massless particles
16.4.4 Fractional electric charges
16.4.5 Discussion
16.5 Global symmetries
16.5.1 CP conservation in superstring models
16.5.2 R transformations in superstring models
16.5.3 Global symmetries of the toy model
16.5.4 Transformation laws of matter fields
16.6 Topological formulas for Yukawa couplings
16.6.1 A topological formula for the superpotential
16.6.2 The kinetic terms
16.6.3 A nonrenormalization theorem and its consequences
16.6.4 Application to the toy model
16.7 Another approach to symmetry breaking
16.8 Discussion
16.9 Renormalization of coupling constants
16.10 Orbifolds and algebraic geometry
16.11 Outlook
Bibliography
Index